The Cayley trick and triangulations of products of simplices
| dc.creator | Santos, Francisco | |
| dc.date | 2003-12-02 | |
| dc.date | 2004-04-08 | |
| dc.date.accessioned | 2026-07-07T05:03:30Z | |
| dc.date.available | 2026-07-07T05:03:30Z | |
| dc.description | We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) $l$, we show that the numbers of regular and non-regular triangulations of $Δ^l\timesΔ^k$ grow, respectively, as $k^{Θ(k)}$ and $2^{Ω(k^2)}$. For the special case of $l=2$, we relate triangulations to certain class of lozenge tilings. This allows us to compute the exact number of triangulations up to $k=15$, show that the number grows as $e^{βk^2/2 + o(k^2)}$ where $β\simeq 0.32309594$ and prove that the set of all triangulations is connected under geometric bistellar flips. The latter has as a corollary that the toric Hilbert scheme of the determinantal ideal of $2\times 2$ minors of a $3\times k$ matrix is connected, for every $k$. We include ``Cayley Trick pictures'' of all the triangulations of $Δ^2\times Δ^2$ and $Δ^2\times Δ^3$, as well as one non-regular triangulation of $Δ^2\times Δ^5$ and one of $Δ^3\times Δ^3$. | |
| dc.description | This version has been accepted in "Proceedings of the Joint Summer Research Conference on Integer Points in Polyhedra" (Barvinok et al., eds.) Contemporary Mathematics, American Mathematical Society. Changes from v2: corrected a LaTeX problem with figures. Changes from v1: (1) some rephrasing, especially in the introduction. (2) the former proof of Theorem 5.4 was incorrect. The bound in the statement has been changed to match the new proof | |
| dc.identifier | https://arxiv.org/abs/math/0312069 | |
| dc.identifier | http://arxiv.org/abs/math/0312069 | |
| dc.identifier | In "Integer Points in Polyhedra - Geometry, Number Theory, Algebra, Optimization", A. Barvinok, M. Beck, C. Haase, B. Reznick, and V. Welker (eds), Contemporary Mathematics 374, Amer. Math. Soc., Providence, 2005. ISBN 0-8218-3459-2. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/69444 | |
| dc.subject | Combinatorics | |
| dc.subject | 52B20; 52B11 | |
| dc.title | The Cayley trick and triangulations of products of simplices | |
| dc.type | text |